A plasma is an ionized gas whose charged particle constituents (electrons and ions) interact primarily through electromagnetic forces. Plasmas in various forms make up a large portion of the known universe. In addition, they form the basis of a multitude of everyday devices from incandescent or fluorescent lighting to recent plasma television development. They are also used in the manufacture and processing of printed circuit boards, have recently been shown to have medical application, and are the focus of research in areas such as power generation associated with nuclear fusion reactors.
To characterize a plasma, various diagnostics must be performed to understand the physics of particle interactions. One of these parameters is plasma potential φp. This is the potential of the plasma at a particular location usually with respect to an experimental chamber wall. This is distinguished from the floating potential at which an object immersed in the plasma collects zero net current. Floating potential is typically negative with respect to plasma potential and the two are generally not equal.
The most widely used technique still today for determining plasma parameters such as the plasma potential φp or the electron energy distribution function ƒ(ε) is through the use of a Langmuir probe placed in the plasma. I. Langmuir and H. Mott Smith, “The theory of collectors in gaseous discharges”, Phys. Rev. 28, 727 (1926).
One method for finding plasma potential φp which involves using a Langmuir probe relies on current collection using a dc voltage sweep, and from these data the dc current characteristic of the plasma Ip(Vp) is derived, where Ip is the probe collected current and Vp is the probe voltage. R. L. Merlino, “Understanding Langmuir Probe current-voltage characteristics,” Am. J. Phys. 75, 1078 (2007).
In Langmuir probe-based methods, plasma potential φp is determined by noting that the probe collected current Ip with respect to probe voltage Vp falls rapidly once the applied probe voltage exceeds the plasma potential φp. Thus,
      ⅆ          I      p            ⅆ          V      p      has a peak, and
                                          ⅆ            2                    ⁢                      I            p                                    ⅆ                      V            p            2                                                    V        p            =              ϕ        p              ≃  0.See Godyak et al., “Probe diagnostics of non-Maxwellian Plasmas,” J. AppL Phys 73, 3657 (1993). Godyak asserts that this method gives an unequivocal value for the plasma potential φp. Id.
Thus, conventional methods of finding plasma potential φp using a Langmuir probe require taking a second derivative of Ip(Vp) and determining the inflection point of Ip(Vp), i.e., the point where
                    ⅆ        2            ⁢              I        p                    ⅆ              V        p        2              =  0.
However, Langmuir probes are susceptible to contamination, and in many cases calculating the second derivative often is severely affected by noise and so introduces errors in the values of φp.
Consequently, to avoid having to calculate the second derivative, many researchers resort to fitting routines of various forms, based in part on the probe geometry, to determine the inflection point, i.e., the point where
                    ⅆ        2            ⁢              I        p                    ⅆ              V        p        2              =  0.See, e.g., J. J. Carroll, et. al., “A segmented disk electrode to produce and control parallel and transverse particle drifts in a cylindrical plasma,” Rev. Sci. Instrum., 65(9), 2991 (1994). These fitting routines also have been used to avoid errors introduced by probe contamination, but by their nature are only approximate and most often assume a Maxwellian distribution. Since the fit itself treats a complete curve, a fit to one area of the curve (such as the electron saturation region) influences the entire curve fit and therefore the determination of plasma parameters. Also fitting routines should be based on physical reasoning and not on the assumption of prevailing geometry (i.e., algebraic fits) as is often the case. R. F. Fernsler, “Modeling Langmuir probes in multi-component plasmas,” Plasma Sources Sci. Technol. 18, 014012 (2009)
Other methods which also attempt to avoid having to make this double differentiation of the current-voltage characteristic use ac voltages in an indirect determination. By superposing on V a small constant ac voltage component, ξ sin ωt , such that ξ<<V, it can be shown that the time-averaged increment to the current gives the second derivative above. Yu. P. Raizer, Gas Discharge Physics, p. 111, (Springer-Verlag, Berlin Heidelberg New York, 1997).
However, noise remains a serious problem in all such cases. Whether calculating a second order derivative, using fitting routines or time averaging, even small amounts of noise can produce large fluctuations which generate uncertainty in the value of φp.
The inventors herein have explored the use of an rf probe to determine parameters such as electron temperature Te, electron density ne(r), and electron sheath profile structure. See D. N. Walker, R. F. Fernsler, D. D. Blackwell, and W. E. Amatucci, “On the Non-intrusive Determination of Electron Density in the Sheath of a Spherical Probe,” Naval Research Laboratory Memorandum Report, NRL/MR/6750-07-9033, Apr. 20, 2007 (“Walker 2007”); D. N. Walker, R. F. Fernsler, D. D. Blackwell, and W. E. Amatucci, “Determining electron temperature for small spherical probes from network analyzer measurements of complex impedance,” Physics of Plasmas 15, 123506 (2008) (“Walker 2008”); see also D. N. Walker, R. F. Fernsler, D. D. Blackwell, W. E. Amatucci, and S. J. Messer, “On collisionless energy absorption in plasmas: Theory and experiment in spherical geometry,” Physics of Plasmas 13, 032108 (2006) (“Walker 2006”), all of which are hereby incorporated by reference into the present application in their entirety.